Electron dynamics inside a vacuum tube diode through linear differential equations

TL;DR

This paper presents a linear differential equation approach to analyze electron motion in vacuum tube diodes, simplifying the classical Child-Langmuir model by avoiding nonlinear equations and providing known current-voltage behavior.

Contribution

It introduces a linear differential equation method for diode analysis, bypassing nonlinear equations and deriving classical current-voltage relationships.

Findings

Reproduces classical current density proportional to bias potential^{3/2}

Shows current inversely proportional to square of electrode gap

Simplifies analysis by avoiding nonlinear differential equations

Abstract

In this paper we analyze the motion of charged particles in a vacuum tube diode by solving linear differential equations. Our analysis is based on expressing the volume charge density as a function of the current density and coordinates only, while in the usual scheme the volume charge density is expressed as a function of the current density and electrostatic potential. Our approach gives the well known behavior of the classical current density proportional to the three-halves power of the bias potential and inversely proportional to the square of the gap distance between the electrodes, and does not require the solution of the nonlinear differential equation normally associated with the Child-Langmuir formulation.